Quantifying method of uncertainty of measured value by close match span calibration of measuring sensor and the apparatus using the same

ABSTRACT

Disclosed herein is a quantifying method of uncertainty of measured value by close match span calibration of measuring sensor comprising a step of:a quantifying using below equation I.uc(xbag)≃[1fcyl⁢u⁡(Rbag)]2+[1fcyl⁢u⁡(Rcyl)]2+u2(xcyl)(I)wherein, uc is a combined standard uncertainty, Xbag is a measured value of a test, xcyl is standard value, fcyl is standard response factor (sensitivity coefficient), Rbag is a signal value of a test, represents xbag·fbag, Rcyl represents xcyl·fcyl, and u is standard uncertainty.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims priority to Korean Patent Application10-2020-0113420, filed on Sep. 4, 2020, the entire disclosure of whichis incorporated herein by reference.

BACKGROUND Field

Embodiments of the present invention relate to quantifying method ofuncertainty of measured value by close match span calibration and theapparatus, and, more particularly, to quantifying method of uncertaintyof measured value by close match span calibration of measuring sensorand the apparatus to provide a method and an apparatus to derive ageneralized equation to quantify the measurement uncertainty using theOPCM calibration, quantifying relative contributions of relevantuncertainty sources.

Description of the Related Art

The uncertainty of measured value (ex, voltage, current, resistance)using sensors are quantitative characteristics of the quality or thecredibility of said sensors.

Up to now, there has been ‘offset and span calibration’ method forcalibration method used to correct said sensors.

Said method uses linear transfer function which is quantitativelyrelated function between measured values and sensor output signals, andneeds the process of correcting offset regarding y-axis to zero. Thiscalibration method is called one-point through origin (OPTO)

The one-point through-origin (OPTO) calibration is being frequently usedfor quantification of analyte in a sample where a calibration line usingstandards must pass through a zero response in instrumentation. Thisrequirement is not valid in many cases.

Recently one-point close-match (OPCM) calibration is briefly introducedto an international standard. (ISO 12963: 2017).

The OPCM calibration model has an advantage over the OPTO calibration inthat the former does not require the assumption of linearity throughorigin.

However, information is basically lacking on how to derive theuncertainty estimation model based on the OPCM calibration and how toquantitatively assess the influence of relevant uncertainty sources.Information with regard to these two has not been reported to date.

Therefore, under these circumstances there are some problems of notbeing able to optimize managements because it is difficult to determinethe priority among managements regarding relevant uncertainty sources.

SUMMARY

Therefore, the present invention has been made in view of the aboveproblems, and it is an object of the present invention to provide amethod and an apparatus to derive a generalized equation to quantify themeasurement uncertainty using the OPCM calibration, quantifying relativecontributions of relevant uncertainty sources.

In accordance with a first aspect of present invention, the aboveobjects can be accomplished by following means;

A quantifying method of uncertainty of measured value by close matchspan calibration of measuring sensor using below equation I.

$\begin{matrix}{{u_{c}\left( x_{bag} \right)} \simeq \sqrt{\left\lbrack {\frac{1}{f_{cyl}}{u\left( R_{bag} \right)}} \right\rbrack^{2} + \left\lbrack {\frac{1}{f_{cyl}}{u\left( R_{cyl} \right)}} \right\rbrack^{2} + {u^{2}\left( x_{cyl} \right)}}} & (I)\end{matrix}$

wherein, u_(c) is a combined standard uncertainty, X_(bag) is a measuredvalue of a test, x_(cyl) is standard value, f_(cyl) is standard responsefactor (sensitivity coefficient), R_(bag) is a signal value of a test,represents x_(bag)·f_(bag), R_(cyl) represents x_(cyl)·f_(cyl), and u isstandard uncertainty.

In accordance with the first aspect of the present invention, the aboveand other objects can be accomplished by quantifying relativecontributions of relevant uncertainty sources using below equation II.

$\begin{matrix}{{{h\left( R_{bag} \right)} = \frac{\left\lbrack {\frac{1}{f_{cyl}}{u\left( R_{bag} \right)}} \right\rbrack^{2}}{u_{c}^{2}\left( x_{bag} \right)}};} & ({II})\end{matrix}$${{h\left( R_{cyl} \right)} = \frac{\left\lbrack {\frac{1}{f_{cyl}}{u\left( R_{cyl} \right)}} \right\rbrack^{2}}{u_{c}^{2}\left( x_{bag} \right)}};$${h\left( x_{cyl} \right)} = \frac{u^{2}\left( x_{cyl} \right)}{u_{c}^{2}\left( x_{bag} \right)}$

In accordance with the first aspect of the present invention, preferablythe measuring sensor is gas sensor.

In accordance with a second aspect of the present invention, the aboveobjects can be accomplished by a quantifying apparatus of uncertainty ofmeasured value by close match span calibration of measuring sensor usingbelow equation I.

$\begin{matrix}{{u_{c}\left( x_{bag} \right)} \simeq \sqrt{\left\lbrack {\frac{1}{f_{cyl}}{u\left( R_{bag} \right)}} \right\rbrack^{2} + \left\lbrack {\frac{1}{f_{cyl}}{u\left( R_{cyl} \right)}} \right\rbrack^{2} + {u^{2}\left( x_{cyl} \right)}}} & (I)\end{matrix}$

wherein, u_(c) is a combined standard uncertainty, X_(bag) is a measuredvalue of a test, x_(cyl) is standard value, f_(cyl) is standard responsefactor (sensitivity coefficient), R_(bag) is a signal value of a test,represents x_(bag)·f_(bag), R_(cyl) represents x_(cyl)·f_(cyl), and u isstandard uncertainty.

In accordance with the second aspect of the present invention, the aboveand other objects can be accomplished by quantifying relativecontributions of relevant uncertainty sources using below equation II.

$\begin{matrix}{{{h\left( R_{bag} \right)} = \frac{\left\lbrack {\frac{1}{f_{cyl}}{u\left( R_{bag} \right)}} \right\rbrack^{2}}{u_{c}^{2}\left( x_{bag} \right)}};} & ({II})\end{matrix}$${{h\left( R_{cyl} \right)} = \frac{\left\lbrack {\frac{1}{f_{cyl}}{u\left( R_{cyl} \right)}} \right\rbrack^{2}}{u_{c}^{2}\left( x_{bag} \right)}};$${h\left( x_{cyl} \right)} = \frac{u^{2}\left( x_{cyl} \right)}{u_{c}^{2}\left( x_{bag} \right)}$

In accordance with the second aspect of the present invention,preferably the measuring sensor is gas sensor.

In accordance with a third aspect of the present invention, the aboveobjects can be accomplished by computer-readable recording mediumrecording program to implement the method.

BRIEF DESCRIPTION OF THE DRAWINGS

The above and other objects, features and other advantages of thepresent invention will be more clearly understood from the followingdetailed description taken in conjunction with the accompanyingdrawings, in which:

FIG. 1 is a block diagram illustrating the construction of quantifyingapparatus of uncertainty of measured value by close match spancalibration.

DETAILED DESCRIPTION OF EMBODIMENTS

Hereinafter, various embodiments according to the present invention willbe described in detail with reference to the accompanying drawings. Thedetailed description to be described below with reference to theaccompanying drawings is intended to illustrate exemplary embodiments ofthe invention and is not intended to represent the only embodiment inwhich the invention may be executed. The following detailed descriptionincludes specific details in order to provide a complete understandingof the present invention. However, those skilled in the art willappreciate that the present invention may be executed without thesespecific details.

In some cases, well-known structures and devices will not be describedor will be illustrated in a block diagram form centering on corefunctions of each structure and apparatus, to avoid obscuring conceptsof the present invention.

In the specification, when the explanatory phrase a part “comprises orincludes” a component is used, this means that the part may furtherinclude the component without excluding other components, so long asspecial explanation is not given. Further, the term “ . . . unit”described in the specification means a unit for processing at least onefunction or operation. In addition, as used herein the context fordescribing the present invention (particularly, in the context of thefollowing claims), the singular forms “a,” “an,” “one” and “the” areintended to include the plural forms as well, unless the context clearlyindicates otherwise in the specification or is clearly limited by thecontext.

In description of exemplary embodiments of the present invention, thepublicly known functions and configurations that are judged to be ableto make the purport of the present invention unnecessarily obscure willnot be described in detail.

Further, wordings to be described below are defined in consideration ofthe functions of the present invention, and may differ depending on theintentions of a user or an operator or custom. Accordingly, suchwordings should be defined on the basis of the contents of the overallspecification.

Hereinafter, exemplary embodiments of the present invention will bedescribed with reference to the accompanying drawings.

Embodiments of the present invention of quantifying method ofuncertainty of measured value by close match span calibration ofmeasuring sensor concern calibrating method, and several sensors such asgas sensor, pressure sensor, current sensor, etc.

Embodiments of the present invention provide means to quantify andcalculate combined standard uncertainty regarding measured values usingOPCM calibration, and quantify relative contributions of relevantuncertainty sources regarding said combined standard uncertainty.

According to exemplary embodiments of the present invention, preferably,above means automatically give input values mechanically, or is giveninput values manually using below equation I to calculate combinedstandard uncertainty automatically.

$\begin{matrix}{{u_{c}\left( x_{bag} \right)} \simeq \sqrt{\left\lbrack {\frac{1}{f_{cyl}}{u\left( R_{bag} \right)}} \right\rbrack^{2} + \left\lbrack {\frac{1}{f_{cyl}}{u\left( R_{cyl} \right)}} \right\rbrack^{2} + {u^{2}\left( x_{cyl} \right)}}} & (I)\end{matrix}$

wherein, u_(c) is a combined standard uncertainty, X_(bag) is measuredvalue of a test, x_(cyl) is standard value, f_(cyl) is standard responsefactor (sensitivity coefficient), R_(bag) is signal value of a test,represents X_(bag)·f_(bag), R_(cyl) represents x_(cyl)·f_(cyl), and u isstandard uncertainty.

Also, according to exemplary embodiments of the present invention, it ispossible to calculate relative contributions of relevant uncertaintysources regarding combined standard uncertainty. Using this result, itis possible to calculate relative contributions of relevant uncertaintysources to contribute to the combined standard uncertainty regardingmeasured results using OPCM calibration.

$\begin{matrix}{{{h\left( R_{bag} \right)} = \frac{\left\lbrack {\frac{1}{f_{cyl}}{u\left( R_{bag} \right)}} \right\rbrack^{2}}{u_{c}^{2}\left( x_{bag} \right)}};} & ({II})\end{matrix}$${{h\left( R_{cyl} \right)} = \frac{\left\lbrack {\frac{1}{f_{cyl}}{u\left( R_{cyl} \right)}} \right\rbrack^{2}}{u_{c}^{2}\left( x_{bag} \right)}};$${h\left( x_{cyl} \right)} = \frac{u^{2}\left( x_{cyl} \right)}{u_{c}^{2}\left( x_{bag} \right)}$

As described above, if we get relative contributions of relevantuncertainty sources h(R_(bag)), h(R_(cyl)), and h(X_(cyl)) usingequation II, it is possible to use as useful information to determinethe priority among managements regarding relevant uncertainty sources.

Hereinafter, exemplary embodiments of the present invention will bedescribed in detail regarding the process to derive related function forevaluation model of combined standard uncertainty, and calculation ofrelative contributions of relevant uncertainty sources to contribute tothe combined standard uncertainty. In these embodiments, the measuringsensor is gas sensor, and the gas is methane as one of green house gasspecies

A requirement for the OPCM calibration is that, for example, the amountfraction of methane, a greenhouse gas species, in a calibration cylindermust be very close to that in the sampling bag:X _(bag)=(X _(bag) −x _(cyl))+X _(cyl)  (1)

The response factors (i.e., instrument sensitivity) of methane in thesampling bag and the calibration cylinder using a typicalinstrumentation (e.g., GC-FID) are defined as:

$\begin{matrix}{{f_{bag} = \frac{R_{bag}}{x_{bag}}};} & (2)\end{matrix}$ $f_{cyl} = \frac{R_{cyl}}{x_{cyl}}$

The methane fractions in the sampling bag and the calibration gascylinder are expressed as a function of the response factor f (i.e.,calibration function), respectively.

$\begin{matrix}{{x_{bag} = \frac{R_{bag}}{f_{bag}}};} & (3)\end{matrix}$ $x_{cyl} = \frac{R_{cyl}}{f_{cyl}}$

This is applied to the right-hand side of Equation (1), which yields:

$\begin{matrix}{x_{bag} = {\left( {\frac{R_{bag}}{f_{bag}} - \frac{R_{cyl}}{f_{cyl}}} \right) + x_{cyl}}} & (4)\end{matrix}$

The response factor f of the instrument should be constant to the samecomponent methane under the close-match conditions regardless of the gascontainers used:f _(bag) =f _(cyl)  (5)

Now Equation (4) can be expressed as:

$\begin{matrix}{x_{bag} = {{\frac{1}{f_{cyl}}\left( {R_{bag} - R_{cyl}} \right)} + x_{cyl}}} & (6)\end{matrix}$

Therefore the combined standard uncertainty of the methane fraction inthe sampling bag:

$\begin{matrix}{{u_{c}\left( x_{bag} \right)} = {u\left\lbrack {{\frac{1}{f_{cyl}}\left( {R_{bag} - R_{cyl}} \right)} + x_{cyl}} \right\rbrack}} & (7)\end{matrix}$

Squaring both sides yields:

$\begin{matrix}{{u_{c}^{2}\left( x_{bag} \right)} = {u^{2}\left\lbrack {{\frac{1}{f_{cyl}}\left( {R_{bag} - R_{cyl}} \right)} + x_{cyl}} \right\rbrack}} & (8)\end{matrix}$

The law of uncertainty propagation with a 1st order Taylor series isapplied to Equation (8) in order to derive the square of the combinedstandard uncertainty of methane in a sampling bag:

$\begin{matrix}{{u_{c}^{2}\left( x_{bag} \right)} = {{u^{2}\left\lbrack {\frac{1}{f_{cyl}}\left( {R_{bag} - R_{cyl}} \right)} \right\rbrack} + {u^{2}\left( x_{cyl} \right)}}} & (9)\end{matrix}$

where u(xcyl) is the combined standard uncertainty of the methanefraction in the calibration cylinder.

Multiplying

$\left\lbrack {\frac{1}{f_{cyl}}\left( {R_{bag} - R_{cyl}} \right)} \right\rbrack^{2},$and dividing by

$\left\lbrack {\frac{1}{f_{cyl}}\left( {R_{bag} - R_{cyl}} \right.} \right\rbrack^{2}$for the righthand side of Equation (9) yields:

$\begin{matrix}{{u_{c}^{2}\left( x_{bag} \right)} = {{\left\lbrack {\frac{1}{f}\left( {R_{bag} - R_{cyl}} \right)} \right\rbrack^{2}\frac{u^{2}\left\lbrack {\frac{1}{f_{cyl}}\left( {R_{bag} - R_{cyl}} \right)} \right\rbrack}{\left\lbrack {\frac{1}{f_{cyl}}\left( {R_{bag} - R_{cyl}} \right)} \right\rbrack^{2}}} + {u^{2}\left( x_{cyl} \right)}}} & (10)\end{matrix}$ $\begin{matrix}{{u_{c}^{2}\left( x_{bag} \right)} = {{\left\lbrack {\frac{1}{f_{cyl}}\left( {R_{bag} - R_{cyl}} \right)} \right\rbrack^{2}{u_{rel}^{2}\left\lbrack {\frac{1}{f_{cyl}}\left( {R_{bag} - R_{cyl}} \right)} \right\rbrack}} + {u^{2}\left( x_{cyl} \right)}}} & (11)\end{matrix}$ where, $\begin{matrix}{{u_{rel}^{2}\left\lbrack {\frac{1}{f_{cyl}}\left( {R_{bag} - R_{cyl}} \right)} \right\rbrack} = \left\lbrack {{u_{rel}^{2}(f)} + {u_{rel}^{2}\left( {R_{bag} - R_{cyl}} \right)}} \right\rbrack} & (12)\end{matrix}$

Now using Equation (12), Equation (11) can be expressed as:

$\begin{matrix}{{u_{c}^{2}\left( x_{bag} \right)} = {{\left\lbrack {\frac{1}{f_{cyl}}\left( {R_{bag} - R_{cyl}} \right)} \right\rbrack^{2}\left\lbrack {{u_{rel}^{2}(f)} + {u_{rel}^{2}\left( {R_{bag} - R_{cyl}} \right)}} \right\rbrack} + {u^{2}\left( x_{cyl} \right)}}} & (13)\end{matrix}$ $\begin{matrix}{{u_{c}^{2}\left( x_{bag} \right)} = {{\left\lbrack {\frac{1}{f_{cyl}}\left( {R_{bag} - R_{cyl}} \right)} \right\rbrack^{2}\left\lbrack {\frac{y^{2}\left( f_{cyl} \right)}{f_{{cyl}^{2}}} + \frac{u^{2}\left( {R_{bag} - R_{cyl}} \right)}{\left( {R_{bag} - R_{cyl}} \right)^{2}}} \right\rbrack} + {u^{2}\left( x_{cyl} \right)}}} & (14)\end{matrix}$

Applying the law of uncertainty propagation for addition or subtractionto u²(R_(bag)−R_(cyl)), yields:u ²(R _(bag) −R _(cyl))=u ²(R _(bag))+u ²(R _(cyl))  (15)

Thus Equation (14) would be:

$\begin{matrix}{{u_{c}^{2}\left( x_{bag} \right)} = {{\left( \frac{1}{f_{cyl}} \right)^{2}{\left( {R_{bag} - R_{cyl}} \right)^{2}\left\lbrack {\frac{u^{2}\left( f_{cyl} \right)}{f_{{cyl}^{2}}} + \frac{{u^{2}\left( R_{bag} \right)} + {u^{2}\left( R_{cyl} \right)}}{\left( {R_{bag} - R_{cyl}} \right)^{2}}} \right\rbrack}} + {u^{2}\left( x_{cyl} \right)}}} & (16)\end{matrix}$ $\begin{matrix}{{u_{c}^{2}\left( x_{bag} \right)} = {{\frac{u_{(f)}^{2}}{f_{{cyl}^{4}}}\left( {R_{bag} - R_{cyl}} \right)^{2}} + {\frac{1}{f_{{cyl}^{2}}}\left\lbrack {{n^{2}\left( R_{bag} \right)} + {u^{2}\left( R_{cyl} \right)}} \right\rbrack} + {u^{2}\left( x_{cyl} \right)}}} & (17)\end{matrix}$ $\begin{matrix}{{u_{c}\left( x_{bag} \right)} = \sqrt{{\frac{u^{2}\left( f_{cyl} \right)}{f_{cyl}^{4}}\left( {R_{bag} - R_{cyl}} \right)^{2}} + {\frac{1}{f_{cyl}^{2}}\left\{ {{u^{2}\left( R_{bag} \right)} + {u^{2}\left( R_{cyl} \right)}} \right\}} + {u^{2}\left( x_{cyl} \right)}}} & (18)\end{matrix}$ $\begin{matrix}{{u_{c}\left( x_{bag} \right)} = \sqrt{\left\lbrack {\frac{\left( {R_{bag} - R_{cyl}} \right)}{f_{cyl}^{2}}{u\left( f_{cyl} \right)}} \right\rbrack^{2} + \left\lbrack {\frac{1}{f_{cyl}}{u\left( R_{bag} \right)}} \right\rbrack^{2} + \left\lbrack {\frac{1}{f_{cyl}}{u\left( R_{cyl} \right)}} \right\rbrack^{2} + {u^{2}\left( x_{cyl} \right)}}} & (19)\end{matrix}$ $\begin{matrix}{{where},{{u\left( f_{cyl} \right)} = {u\left( \frac{R_{cyl}}{x_{cyl}} \right)}}} & (20)\end{matrix}$

Applying the law of uncertainty propagation for multiplication ordivision yields:

$\begin{matrix}{\left\lbrack \frac{u\left( f_{cyl} \right)}{f_{cyl}} \right\rbrack^{2} = {\left\lbrack \frac{u\left( R_{cyl} \right)}{R_{cyl}} \right\rbrack^{2} + \left\lbrack \frac{u\left( x_{cyl} \right)}{x_{cyl}} \right\rbrack^{2}}} & (21)\end{matrix}$

Using Equation (21), Equation (19) can be expressed as:

$\begin{matrix}{{u_{c}\left( x_{bag} \right)} = \sqrt{\begin{matrix}{\left\lbrack {\frac{\left( {R_{bag} - R_{cyl}} \right)}{f_{cyl}}\sqrt{\left\lbrack {\frac{u^{2}\left( R_{cyl} \right)}{R_{{cyl}^{2}}} + \frac{u^{2}\left( x_{cyl} \right)}{x_{{cyl}^{2}}}} \right\rbrack}} \right\rbrack^{2} +} \\{\left\lbrack {\frac{1}{f_{cyl}}u\left( R_{bag} \right)} \right\rbrack^{2} + \left\lbrack {\frac{1}{f_{cyl}}{u\left( R_{cyl} \right)}} \right\rbrack^{2} + {u^{2}\left( x_{cyl} \right)}}\end{matrix}}} & (22)\end{matrix}$

Now that the requirement must be met (i.e., is very near to):

$\begin{matrix}{\left\lbrack {\frac{\left( {R_{bag} - R_{cyl}} \right)}{f_{cyl}}\sqrt{\left\lbrack {\frac{u^{2}\left( R_{cyl} \right)}{R_{{cyl}^{2}}} + \frac{u^{2}\left( x_{cyl} \right)}{x_{{cyl}^{2}}}} \right\rbrack}} \right\rbrack^{2}{{\left\lbrack {\frac{1}{f_{cyl}}{u\left( R_{bag} \right)}} \right\rbrack^{2} +}}} & (23)\end{matrix}$$\left. {\left\lbrack {\frac{1}{f_{cyl}}u\left( R_{cyl} \right)} \right\rbrack^{2} + {u^{2}\left( x_{cyl} \right.}} \right)$

As a consequence, a simplified equation in terms of variance is derived:

$\begin{matrix}{{u_{c}\left( x_{bag} \right)} \simeq \sqrt{\left\lbrack {\frac{1}{f_{cyl}}{u\left( R_{bag} \right)}} \right\rbrack^{2} + \left\lbrack {\frac{1}{f_{cyl}}{u\left( R_{cyl} \right)}} \right\rbrack^{2} + {u^{2}\left( x_{cyl} \right)}}} & (I)\end{matrix}$

wherein, u_(c) is a combined standard uncertainty, X_(bag) is a measuredvalue of a test, x_(cyl) is standard value, f_(cyl) is standard responsefactor (sensitivity coefficient), R_(bag) is a signal value of a test,represents x_(bag)·f_(bag), R_(cyl) represents x_(cyl)·f_(cyl), and u isstandard uncertainty.

The relative contribution (%) of each uncertainty source on the combinedstandard uncertainty of is estimated:

$\begin{matrix}{{{h\left( R_{bag} \right)} = \frac{\left\lbrack {\frac{1}{f_{cyl}}{u\left( R_{bag} \right)}} \right\rbrack^{2}}{u_{c}^{2}\left( x_{bag} \right)}};} & ({II})\end{matrix}$${{h\left( R_{cyl} \right)} = \frac{\left\lbrack {\frac{1}{f_{cyl}}{u\left( R_{cyl} \right)}} \right\rbrack^{2}}{u_{c}^{2}\left( x_{bag} \right)}};$${h\left( x_{cyl} \right)} = \frac{u^{2}\left( x_{cyl} \right)}{u_{c}^{2}\left( x_{bag} \right)}$

In addition, exemplary embodiments of the present invention, as shown inFIG. 1 , provide an apparatus 100 for quantifying uncertainty ofmeasured value by close match span calibration of measuring sensorcomprising a means for quantifying uncertainty 30, and a means forassaying relative contributions of relevant uncertainty sourcesregarding said uncertainty 50.

More specifically, the apparatus 100 comprises an inputting means 10which inputs data of f_(cyl), u(R_(bag)), u(R_(cyl)), and u(x_(cyl)), aquantifying means 30 which calculates combined standard uncertaintyu_(c)(x_(bag)) using equation I after receiving said data, an evaluatingmeans 50 for evaluating relative contributions of relevant uncertaintysources regarding said uncertainty using equation II, an outputtingmeans 70 which outputs the results calculated by an evaluating means 50.

$\begin{matrix}{{u_{c}\left( x_{bag} \right)} \simeq \sqrt{\left\lbrack {\frac{1}{f_{cyl}}{u\left( R_{bag} \right)}} \right\rbrack^{2} + \left\lbrack {\frac{1}{f_{cyl}}{u\left( R_{cyl} \right)}} \right\rbrack^{2} + {u^{2}\left( x_{cyl} \right)}}} & (I)\end{matrix}$

wherein, u_(c) is a combined standard uncertainty, X_(bag) is a measuredvalue of a test, x_(c)y_(j) is standard value, f_(cyl) is standardresponse factor (sensitivity coefficient), R_(bag) is a signal value ofa test, represents x_(bag)·f_(bag), R_(cyl) represents x_(cyl)·f_(cyl),and u is standard uncertainty.

$\begin{matrix}{{{h\left( R_{bag} \right)} = \frac{\left\lbrack {\frac{1}{f_{cyl}}{u\left( R_{bag} \right)}} \right\rbrack^{2}}{u_{c}^{2}\left( x_{bag} \right)}};} & ({II})\end{matrix}$${{h\left( R_{cyl} \right)} = \frac{\left\lbrack {\frac{1}{f_{cyl}}{u\left( R_{cyl} \right)}} \right\rbrack^{2}}{u_{c}^{2}\left( x_{bag} \right)}};$${h\left( x_{cyl} \right)} = \frac{u^{2}\left( x_{cyl} \right)}{u_{c}^{2}\left( x_{bag} \right)}$

The quantifying means 30 and the evaluating means 50 can be implementedto computer-readable program, optionally can be implemented usingfirmware or hardware, and Micom can be used to calculate equation I andequation II.

According to exemplary embodiments of the present invention, asimplified equation for the uncertainty estimation model of the OPCMcalibration is newly developed using the law of uncertainty propagation(1st-order Taylor series model). This model can assess the relativecontributions of relevant uncertainty sources.

An important application of this uncertainty model needs to behighlighted: an analyst can determine whether observed biases of manualor automated gas sampling methods used in the GC-FID are statisticallysignificant. This uncertainty model would be widely applicable toone-point calibration in analytical sciences (e.g., chemical analysis)where it is impossible in reality to assume linearity through origin.

A limited number of possible embodiments for the present teachings havebeen presented above for illustrative purposes. Those of ordinary skillin the art will appreciate that various modifications, additions, andsubstitutions are possible. While this patent document contains manyspecifics, these should not be construed as limitations on the scope ofthe present teachings or of what may be claimed, but rather asdescriptions of features that may be specific to particular embodiments.Certain features that are described in this patent document in thecontext of separate embodiments can also be implemented in combinationin a single embodiment. Conversely, various features that are describedin the context of a single embodiment can also be implemented inmultiple embodiments separately or in any suitable subcombination.Moreover, although features may be described above as acting in certaincombinations and even initially claimed as such, one or more featuresfrom a claimed combination can in some cases be excised from thecombination, and the claimed combination may be directed to asubcombination or variation of a subcombination.

What is claimed is:
 1. A quantifying method of uncertainty of a measuredvalue by close match span calibration of a gas sensor comprising thesteps of: taking the measured value from the gas sensor; and quantifyingthe uncertainty of the measured value, using below equation I,$\begin{matrix}{{u_{c}\left( x_{bag} \right)} \simeq \sqrt{\left\lbrack {\frac{1}{f_{cyl}}{u\left( R_{bag} \right)}} \right\rbrack^{2} + \left\lbrack {\frac{1}{f_{cyl}}{u\left( R_{cyl} \right)}} \right\rbrack^{2} + {u^{2}\left( x_{cyl} \right)}}} & (I)\end{matrix}$ wherein, u_(c) is a combined standard uncertainty, X_(bag)is a measured value of a test of a sampling bag, taken from the gassensor, x_(cyl) is a standard value, f_(bag) is a response factor of thetest of the sampling bag, f_(cyl) is a standard response factor (asensitivity coefficient), R_(bag) is a signal value of a test, which isx_(bag)·f_(bag), R_(cyl) is a standard signal value of the test, whichis x_(cyl)·f_(cyl), and u is standard uncertainty.
 2. The quantifyingmethod according to claim 1, further comprising the step of: quantifyingrelative contributions of relevant uncertainty sources using belowequation II $\begin{matrix}{{{h\left( R_{bag} \right)} = \frac{\left\lbrack {\frac{1}{f_{cyl}}{u\left( R_{bag} \right)}} \right\rbrack^{2}}{u_{c}^{2}\left( x_{bag} \right)}};{{h\left( R_{cyl} \right)} = \frac{\left\lbrack {\frac{1}{f_{cyl}}{u\left( R_{cyl} \right)}} \right\rbrack^{2}}{u_{c}^{2}\left( x_{bag} \right)}};{{h\left( x_{cyl} \right)} = \frac{u^{2}\left( x_{cyl} \right)}{u_{c}^{2}\left( x_{bag} \right)}}} & ({II})\end{matrix}$
 3. A quantifying apparatus of uncertainty of a measuredvalue by close match span calibration of a gas sensor, wherein thequantifying apparatus is configured to take the measured value from thegas sensor and to use below equation I, $\begin{matrix}{{u_{c}\left( x_{bag} \right)} \simeq \sqrt{\left\lbrack {\frac{1}{f_{cyl}}{u\left( R_{bag} \right)}} \right\rbrack^{2} + \left\lbrack {\frac{1}{f_{cyl}}{u\left( R_{cyl} \right)}} \right\rbrack^{2} + {u^{2}\left( x_{cyl} \right)}}} & (I)\end{matrix}$ wherein, u_(c) is a combined standard uncertainty, X_(bag)is a measured value of a test of a sampling bag, taken from the gassensor, x_(cyl) is a standard value, f_(cyl) is a standard responsefactor (a sensitivity coefficient), R_(bag) is a signal value of a test,which is x_(bag)·f_(bag), R_(cyl) is a standard signal value of thetest, which is x_(cyl)·f_(cyl), and u is standard uncertainty.
 4. Thequantifying apparatus according to claim 3, further comprising:quantifying means of relative contributions of relevant uncertaintysources using below equation II $\begin{matrix}{{{h\left( R_{bag} \right)} = \frac{\left\lbrack {\frac{1}{f_{cyl}}{u\left( R_{bag} \right)}} \right\rbrack^{2}}{u_{c}^{2}\left( x_{bag} \right)}};{{h\left( R_{cyl} \right)} = \frac{\left\lbrack {\frac{1}{f_{cyl}}{u\left( R_{cyl} \right)}} \right\rbrack^{2}}{u_{c}^{2}\left( x_{bag} \right)}};{{h\left( x_{cyl} \right)} = \frac{u^{2}\left( x_{cyl} \right)}{u_{c}^{2}\left( x_{bag} \right)}}} & ({II})\end{matrix}$